\(N\)-body choreographies are periodic solutions to the N-body equations in which
N equal masses chase each other around a fixed closed curve, equally spaced in phase along the
curve. See eq (1) below for the formal choreography condition.
Impetus for their study was born out of the
Chenciner and Montgomery, 2000 rediscovery and existence proof of the (Moore 1994)
figure eight solution of the three-body problem.
.

Carlés Simó, 2000 coined the term "choreography" and found
an astounding number. Here are a few:

The simplest choreographies are found amongst the planar relative equilibria. Place N equal masses at the vertices of a regular N-gon of radius \(a\ .\)
Spin the N-gon about its center
at the correct angular speed \(\omega\)
(with \(\omega^3 a^2 = \mathrm{const}.\) according to the third of [Kepler's laws])
to obtain a solution. For three masses we get a special case of Lagrange's equilateral solutions.

Every choreography curve is analytic. Outside of the relative equilibria no choreography curves are believed to be algebraic.

There is strong evidence that the eight is stable in the sense of K.A.M. Kolmogorov-Arnold-Moser;
[1] , provided perturbations are restricted
to be planar and have zero angular momentum.
(Sense can be made of stability in the case of non-zero angular momentum and
non-planar perturbations, by talking about relative stability
and relative choreographies. Establishing this weaker, more general type of stability
is a subtler issue.) The eight is the only choreography known so far which looks to be K.A.M. stable.

Let \(k, \ell\) be positive integers and
take any \((k, \ell)\) Lissajous figure. Joseph Gerver conjectured, in the winter
of 1999 immediately after the rediscovery of the figure eight was announced,
that there is a choreography solution of the \( N\)-body problem,
\( N \ge k + \ell\) whose choreography curve is isotopic to the given Lissajous figure. The eight is close to a standard \((1,2)\) Lissajous figure.
Within a week Gerver substantiated his conjecture for the case \((1,3)\ ,\) by finding a solution to the 4-body problem now known as Gerver's super-eight. (Select `Gerver' in the right panel in the
applet above.)
Soon afterwards Chenciner, Gerver, Montgomery, and Simó, 2000
numerically established the existence of a vast number of Lissajous-like choreography solutions, and a large number of choreographies which were not especially Lissajou-like. Pictures can be found here Simó. Gerver's conjecture
has since been refined by imposing the constraints
that \((k, l)\) be relatively prime, and imposing further constraints
on \(N\) relative to \( k, \ell \ .\)
In the case \((k,l) = (1,2) \) of the eight,
the existence of solutions with dihedral symmetry for any
\(N \ge 3 \) odd is established in Ferrario and Terracini, 2003.
A rigorous computer assisted proof for the existence of Gerver's super-eight is found in Kapela and Zgliczynski 2003.
For many other choreographies rigorous existence
proofs are still lacking.
Choreographies of n equal masses are in fact quite common: for example, they are dense in the Lyapunov families of quasi-periodic solutions in \(\R^3\) which stem from relative equilibria of a regular n-gon, Chenciner and Féjoz, 2009.

The rest of this article is divided into motivations, methods, and further developments.

Motivations

Theoretical.

The N-body problem, \(N > 2\ ,\)

cannot be solved explicitly. Knowledge of particular solutions such as choreographies can be extremely
valuable and may govern the flow in regions of phase space.
This perspective was emphasized by a famous sentence of Poincaré's
"ce qui nous rend ces solutions périodiques si précieuses, c'est qu'elles sont,
pour ainsi dire, la seule brèche par où nous puissions essayer de pénétrer dans une place jusqu'ici réputée inabordable"
("What makes these periodic solutions so precious is that they are, so to say, the only breach we can use
to penetrate into a region reputed to be unapproachable")
- from the end of paragraph 36 of the first volume of his Méthodes Nouvelles de Mecanique Celeste.

Astronomical.

Are there any figure eight orbits in the universe? Douglas Heggie, 2000 together with Piotr Hut performed numerical experiments by throwing binary pairs, all masses equal,

at each other and seeing how often figure eight solutions formed.
On the basis of these numerical experiments and other considerations, they estimate
the number of figure eight orbits in the observable universe
lies in the range of 1 to 100. One does not expect to see any of the other choreographies
due to their dynamical instability. Thus choreographies are of extremely limited interest in
astronomy.

Aesthetics.

Mathematics is as much art as science. The beauty of choreography solutions has captured a number of researchers.

Methods

Overview
The figure eight and subsequent choreographies were found by combining variational methods
with symmetry methods and numerical methods. The action principle
asserts that paths which minimize the classical action subject to certain constraints
are solutions to Newton's equations. C. Moore found the eight by
fixing the braid type of the three body motion, adding additional symmetry constraints,
and then numerically minimizing the action subject to these constraints. Chenciner
and Montgomery rediscovered the eight by minimizing the classical action amongst all paths which connect an Euler configuration with
mass 2 in the middle to an Isosceles configuration with mass 3 being the symmetry vertex.
Numerical investigations by Simó proved crucial in trusting and understanding
these 1st variational results, and later on, for establishing the existence
of infinite families of choreographies.

Variational methods.

The Principle of least action rewrites Newton's equations as a critical point condition for the classical action \(A\ .\) We closely follow the notation of three body, the `equations' section. \(A(q) = \int_0 ^T L(q(t), \dot q(t)) dt\) is a function of paths \(q(t)\) in the configuration space \(\mathcal X\ .\) \(L(q(t), \dot q(t)) = K(\dot q (t)) + U(q(t))\) is the Lagrangian, the sum of the kinetic energy \(K\) and the negative \(U\) of the potential energy. (The total energy is \(H = K - U \)) If we write \(q_i (t) \in \mathbb R^d \) for the position of the \(i\)th mass

Write \(dA (q) (\delta q)\) for the derivative of \(A\)
at the path \(q\ ,\) and in the direction \(\delta q\ .\)
(In the physics literature one finds the notation \(\int \frac{\delta A} {\delta q} \delta q\) for \(dA (q) (\delta q)\ .\))
The direction \(\delta q\ ,\) also known as a variation of \(q\ ,\) is itself a path lying in some linear space of paths.
The action principle asserts that \(q(t)\)
satisfies Newton's equations if and only if \(dA (q) (\delta q) = 0\) for all "admissible"
variations \(\delta q\ .\) Like all principles this principle is not a precise theorem.
The art to turning the action principle into a theorem is largely the art of choosing a good path space \({\mathcal P}\) for \(q\) to vary in.
The ``admissible' variations \(\delta q\) will then be in the tangent space to \({\mathcal P}\ .\)
In the N-body problem most of the technical difficulties in using the principle of least action arise out of the poles of \(U \)
which is to say from collisions of the bodies.

The direct method of the calculus of variations finds critical points of \(A\) by minimization.
The direct method proceeds
as follows

Step 4. Apply standard calculus to conclude that \(q_*\) is a critical point of \(A\) restricted to \({\mathcal P}\) and hence is a solution to Newton's equations, satisfying whatever auxiliary conditions are imposed by \({\mathcal P}\ .\)

In the applications to choreographies, the path space is defined by imposing symmetry conditions on paths.

Symmetries.

where the particle indices \(i\) are taken mod \(N\ :\) \(q_{N+1} = q_1\ .\)
Condition (1) expresses the invariance of the path \(q(t)\) under an action of the
cyclic group \(\Z_N\) on the full \(T\)-periodic path space. The generator of this group acts by cyclically permuting the mass labels
while simultaneously rotating the "time circle" \(\mathbb R/ T \mathbb Z\) by \(1/N\)th of a full revolution.

Newton designed his equations to be invariant under the group of Galilean transformations.
By a standard trick of classical mechanics, we can fix the center of mass of
the system to be zero for all time which reduces the Galilean symmetry group to the product of the orthogonal group \(O(d)\)
of the Euclidean space \(\mathbb R^d\) in which the masses move
and the group \(Iso(\mathbb R)\) of time translations and reflections.
When all \(N\) masses are equal, as for choreographies,
we have the additional symmetry group \(S_N\) of permutations of the masses.
Thus, following notation introduced by Ferrario and Terracini, 2004, the
symmetry group is \(G(d;N) = O(d) \times Iso(\mathbb R) \times S_N\) with an element
\(g = (R, \tau, \sigma) \in G(d;N)\) acting on a path \(q(t)\) by sending it
to the new path \(\tilde q (t)\) with \(\tilde q_i (t) = R(q_{\sigma^{-1}(i)} ( \tau^{-1} (t))\ .\)
The path \(q(t) \)is invariant under \(g\) if \(q_{\sigma(i)} (\tau (t)) = R(q_i (t))\ .\)
(Compare with the choreography condition above where \(R = Id.\)).

For any subgroup \(\Gamma \subset G(d; N)\) we let
\({\mathcal P}^{\Gamma}\) denote the paths \(q(t)\) which are invariant under
every \(g \in \Gamma\ ,\) and which have finite action. We then apply the direct method of
the calculus of variations with path space \( {\mathcal P} = {\mathcal P}^{\Gamma}\ .\)
When successful, we get a solution \(q_*\) to Newton's equations enjoying symmetry \(\Gamma\ .\)
If \(\Gamma\) contains the choreography group \( \Z_N\ ,\)
then this path is automatically a choreography.

Symmetries of the eight
The symmetry group \(\Gamma\) for the figure eight is a 12 element
group containing the \(N=3\) choreography group \(\Z_3\ .\) This group is the symmetry group of a
regular hexagon, usually denoted \(D_6\ .\) Christian Marchal, 1990 had discovered the importance
of this group in the three-body problem long before the discovery of the figure
eight.

The action of \(D_6\) on paths is perhaps best seen on the shape sphere, a two-sphere whose points represent oriented similarity classes of triangles.

Figure 1: F1

The center of the sphere represents triple collision.

The north and south poles represent the two equilateral triangle configurations: one for each orientation
(or vertex labelling). These points are
labelled `Lag' in the figure because they also represent
the orbits discovered by Lagrange in which the masses forever form an equilateral triangle.
Points along the equator represent collinear triangles– triangles with zero area.
Equally spaced along the equator are the three binary collision points, in which two of the
three masses have collided. The great circle passing through such a collision point and the two Lagrange points represents the set of isosceles triangles whose symmetry axis passes through the non-colliding mass.
These 3 great circles hit the equator orthogonally and with it divide the shape sphere into
12 congruent spherical triangles. This is the same \(12\) of \(|D_6| = 12\ .\) The group \(D_6\) is generated by reflections about these
4 great circles, and it acts freely and transitively on the 12 triangles: there is exactly one element
of the group that takes one triangle to another.

A glance at the figure eight curve drawn in the plane shows it has
4-fold symmetry, being symmetric with respect to reflection in the x and the y axes.
This \(4\) is \(4 = |D_6|/ | \Z_3|\ .\) For the explicit action of \(D_6\) on path space
see Chenciner and Montgomery, 2000.

Avoiding Collisions

A path \(q(t)\) is said to have a collision
if there is a point along the path at which some of the distances \(r_{ij}\) vanish.
Collisions are the main obstacle to successfully applying the direct method
to the N-body problem. The Newtonian action fails to be differentiable at a path with a collision,
and hence at step 3 the method breaks down. (Alternatively, one can define a path space which
does not contain collision paths, but then minimizing sequences in this space can converge to
points outside the space which have collisions.)

There are several ways to insure the minimizing path \(q_*\) has no collision.
The simplest way is to "cheat" by changing the potential. For any potential of the form \(\Sigma m_i m_j / r_{ij}^a\)
with \(a \ge 2\) the action has the property that all paths with collision have infinite action and so action minimizers
are collision-free. This approach was taken by Poincaré (with \(a =2\)) in a beautiful two page paper [REF].
But for the Newtonian potential, collision solutions have finite action, so this approach fails.
Perhaps the simplest way to show collision paths
cannot be minimizers for the Newtonian problem is the test path method. It consists of
two steps. Step A) Find a lower bound \(A_{coll.}\) for the action among all paths with collision, and lying in
the closure of \({\mathcal P}\ .\) Step B) Construct a `test path' \(q \in {\mathcal P}\) which is collision-free
and for which \(A(q) < A_{coll.} \ .\) This method works to establish the existence
of the eight.

The most powerful method to date for avoiding collisions is through the use of Marchal's lemma
(see Chenciner, 2002).
Recall the fixed end point problem in the calculus of variations : fix an initial and a final configuration \(q^i, q^f\)
and consider the space of all paths \({\mathcal P} = {\mathcal P} (q^i, q^f; T)\) which
connect these two in time \(T\ .\) Marchal's lemma asserts that with the Newtonian potential any action minimizer for this problem
has no collisions on the open time interval \((0, T)\ .\)

The method of proof of Marchal's lemma is as important as its statement.
Suppose a path \(q_*\) has an interior collision. Perturb the path by perturbing the masses away from
collision in all possible directions \(\delta q\ .\) Average the resulting perturbed actions over a small sphere of such perturbations.
Show that this average perturbed action is less than \(A(q_*)\ ,\) and conclude from this
that the collision path \(q_*\) is not a minimizer.
At the heart of Marchal's original computation is
the harmonicity of the Newtonian potential, and the fact that the value of a harmonic function at a point is equal to the
average of its values over a sphere centered at that point. (This `equal to its average'
idea was later generalized to an inequality
regarding the average which allowed proofs which worked
for cases where the potential is not harmonic, such as that of the
planar N-body problem.)
Ferrario and Terracini, 2004 proved \(\Gamma-\) equivariant
versions of Marchal's lemma, bringing it to the level of a fine art,
and thereby establishing the existence of
a host of infinite families of choreographies.

Numerical

Numerical methods (computers) played an essential role in finding and understanding choreographies from the beginning.

Moore, 1993 used a numerical
implementation of the direct method of the calculus of variations to discovery the eight.
During the rediscovery period of 1999-2000 Carles Simó helped vanquish doubt as to the validity of one of the complicated existence proofs careful numerical investigations.

The numerical implementation boils down to a gradient search in some finite-dimensional approximation of the path space. One can simply discretize the path at equally spaced values of time. In a very effective approach
one writes the action as a function of the Fourier components of the
choreography curve, and truncates the Fourier series at some finite order.
One can check convergence by increasing the order of truncation and see
the change of the resulting minimizing Fourier coefficients.
If the change lies within some tolerance, call it a victory. For refinements,
See Simó or Moore and Nauenberg.

One of the main advantages of numerically implementing the direct method is that it always converges to ``something. However, this ``something may be outside
of the desired class of curves, or have collisions.
Indeed, it can happen that a desired choreography,
having certain topological conditions which we have imposed simply does not exist.
(See Montgomery [XXX] and Chenciner-Gerver-Montgomery-Sim\' [XXX] for examples.) One
disadvantage of the method is its slow convergence when the
choreographies are close to collision. Another big disadvantage is that its implementation provides no information on the dynamical stability of the choreography.
(Indeed dynamical and variational stability seem to be at odds, generally.)

A more
direct approach to finding choreographies is based on Newton's method of finding zeros,
or fixed points. The problem of looking
for choreographies fits inside the general problem of looking for periodic
solutions, a problem which can be solved by the methods of [g].
This method is at the core of [a],[b],[f] and of
the rigorous existence proof by Kapela-Simó [c] of choreographies
having no geometrical symmetry.

The basic idea of [g] requires an initial guess of the
position and velocity of the \(N\)
bodies at \(t=0\) on a suitable Poincaré section.
For example, a Poincaré section might consist of the condition that
one of the bodies be at the maximal distance on the choreography curve to the centre of
mass at \(t=0\) and with that body located on the \(x\)-axis. After numerically integrating Newton's equations up to a time
\(t=T\ ,\) the period, we require that the bodies return to the initial conditions. This return requirement is a fixed point condition which can be solved by Newton's method. The method requires the differential of the Poincaré map which can be
obtained from the numerical integration of the so-called first variational equation.
See [g] for details. Numerical implementation of the direct method of the calculus of variations typically enter in making the initial guess.
Since Newton's method requires the differential of the Poincaré section, the linear
stability or instability of a found orbit follows as a byproduct of the method.

For
choreographies with complicated paths, it is advisable to proceed beginning with strong force potentials like \( 1/r^\alpha\) with \( \alpha=2\) where we are guaranteed the topological type exists, and then using a continuation method to decrease \alpha from \(\alpha=2\) to
the Newtonian case of \( \alpha=1\ .\) In other words, one numerically implements
the Implicit Function Theorem in order to follow the one-parameter family of paths
parameterized by \(\alpha\ .\) If
during the computations, before reaching
\( \alpha=1\ ,\) the minimal distance between two or more of the bodies during
the period tends to
zero, one has a strong indication that the wanted choreography does not
exist. (This idea is found in the original paper Moore, 1993, implemented there via
the direct method.)

The integration of the equations of
motion must be accurate in this approach, especially strong dynamic instabilities are present. To improve accuracy, the problem can be split into pieces, using so-called
parallel shooting methods, which can be thought of as imposing a number of intermediate
Poincaré sections along the way, with corresponding intermediate implementations of
Newton's method.

Further Developments

Other symmetries, other masses.

In applying the direct method above one imposes symmetries with respect to groups containing the choreography group \(\Z_N\ .\)
By taking groups which
do not contain the choreography group many interesting non-choreographic solutions
have been found.

Perhaps the simplest symmetry requirement is the so-called "Italian symmetry",
the \(\Z_2\)-symmetry which requires the configuration to be symmetric with respect to the center of mass every half-period\[x_i (t+ T/2) = -x_i (t)\ .\] This symmetry
is available for arbitrary masses and arbitrary dimensions.
Italian-symmetric solutions exist, and are particularly interesting in \(\R^3\) because, based on a result by R. Moeckel, Chenciner, 2002 proved that minimizers are never planar, and hence they can not be simply relative equilibria. The first interesting
Italian-symmetric solution of the Newtonian N-body problem was the hip-hop solution of Chenciner and Venturelli, 2000 for 4 equal masses in \(\R^3\ .\)
This solution interpolates between two central configurations: a square and a
regular tetrahedron, rotating as it does so, with the masses at the opposite corners of the square rising and falling accordingly to achieve the tetrahedron.
This paper also contains the 1st rigorous existence proof
of interesting new orbits for the Newtonian N-body problem which employs
the direct method.

Beautiful, highly unstable solutions having the symmetry of any one of the
Platonic solids can be obtained using the direct method.
See [5] or
[6]for these solutions, including one with 60 bodies having
dodecahedral symmetry.

Beyond the Italian-symmetric solutions, other solutions with non-equal masses,
in this case due to Chen are described below (Hill's type retrograde solutions).

Is equality of the masses necessary ?

Do N-body choreographies exist with not all masses are equal?
One conjectures not. Surprisingly, this non-existence result has been proven for only less than 6 bodies (Chenciner, 2001), and only when the equal time spacing requirement is imposed. If the equal time spacing requirement is relaxed, nothing is known, even for three bodies.

Other Potentials.

The direct method combined with symmetries applies to any potential of the form \(\Sigma m_i m_j f( r_{ij})\ .\)

Fujiwara et al solved the problem of which potential leads to a figure eight whose
curve is the algebraic lemniscate. Arriving at their answer led to a number of surprising results
and rediscoveries in classical mechanics.

An especially relevant case is that of a homogeneous potential \[ f(r_{ij})= C_\alpha \frac{m_i m_j}{r_{ij}^\alpha} \ ,\]
the logarithmic potential
\( f(r_{ij})=m_i m_j \log(r_{ij}) \) being a limiting case.
Many choreographies (e.g. the figure eight) can be continued with respect to \( \alpha \)
so as to reach the logarithmic potential and beyond to exponents \(\alpha<0\ .\)
Continuation in \( \alpha \) is a key ingredient in some
of the Numerical Methods [TAG IT] .

The theory of bifurcations with symmetry
applies to many choreographies. One starts with
a \(\Gamma-\)symmetric solution, and then systematically breaks the
symmetry group, following the bifurcating branches of less symmetric
solutions, organizing these branches according to the
lattice of subgroups of the original symmetry group.

To apply the theory of bifurcations with symmetry to the eight, one must expand one's view of the eight as a choreography in two directions. One must
allow for "relative choreographies" as a generalization of the `absolute choreographies discussed above.
And one must view the figure eight as living in space, rather than being constrained to the plane.

A solution is a relative choreography
if, when viewed in some rotating frame, rotating with constant angular speed it is a choreography.
Equivalently, when projected onto shape space, it becomes periodic.
If the frequency of rotation and the period of the orbit are rationally related, then, by traversing a relative choreography enough times, we get a true choreography.

The eight, when viewed in space
has three axes of symmetry, say \(e_1, e_2, e_3\ .\)
We can rotate the eight uniformly about any one of these three axes and find a corresponding family
of relative equilibria labelled \(\Gamma_1, \Gamma_2, \Gamma_3\ ,\)
bifurcating out of the eight, with the angular speed of rotation being the bifurcation parameter. If \(e_3\) is orthogonal to the plane of the
eight, then its relative choreography \(\Gamma_3\) remains planar
for all values of the angular momentum, with one lobe pinching off asymmetrically
to collision as the momentum is increased [Henon]. The axes for the other two families \(\Gamma_1, \Gamma_2\) lie in the plane of the eight, and the corresponding relative equilibria are spatial solutions. One family, \(\Gamma_1\ ,\) can be followed all the way to the Lagrange solution
lying in a plane orthogonal to \(e_2\ ,\) and hence orthogonal to the eight's original plane.
(See Chenciner, Fejoz, and Montgomery for details.)

This \(\Gamma_2\) family, viewed backwards as bifurcating out of Lagrange,
is found in the book of Marchal. Marchal observed that
when Newton's equations are linearized about the equal mass Lagrange solution, one of the
linearized solutions satisfies the \(D_6\) symmetry.

The retrograde solutions of Chen.

Applying the variational method in the case of non equal masses, K.C. Chen obtained generalizations of Hill's retrograde solutions to the

lunar problem. These lunar solutions are solutions to the restricted three-body problem in
which the moon rotates about its planet in the opposite sense that the planet rotates about its sun.
Such orbits realize one of the generators of the homology of shape space for the three body problem.
Conley established the existence of such retrograde orbits for finite non-zero masses,
in the region of parameter space in which one of the masses dominated the other two. Chen's variational approach
applies to a complementary region of
the mass parameter space including the case where the three masses are equal .

Chen follows the direct method as described above. His path space is defined by imposing
two symmetries which do not rely on the masses being equal. Identify \(\mathbb R^2\)
with the space \(\mathbb C\) of complex numbers.
Then these symmetries are \(q_i(t + T) = \operatorname{exp}(i \Phi) q_i(t)\) and \(q_i(t) = \bar q_i (-t)\) where \(T\) is the (relative) period of
the orbit, and the complex conjugation \(q_i \mapsto \bar q_i\) implements reflection about the x-axis in the plane.
The angle \( \Phi\) tells how much the system has rotated in inertial space per relative period \(T \ .\)
In addition to these symmetry constraints he imposes a topological winding number constraint to impose
the retrograde condition.

Acknowledgements

The author is grateful to K.C. Chen, Jacques Féjoz, Charlie McDowell,
Rick Moeckel, and Carles Simó for help with
the animations and Carles Simó and Douglas Heggie and Alain Chenciner for
reading and correcting various drafts.